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First let’s consider the concept of the Spherical Wrist Simplification:

All Wrist joint Z’s intersect at a point

The ‘n Frame’ is offset from this Z’s intersection point at a distance dn (the hand span) along the a vector of the desired solution (3rd column of desired orientation sub-matrix!) which is the z direction of the 3rd wrist joint

this matrix, which contains the joint control angles, is then set equal to a ‘U matrix’ prepared by multiplying the inverse of the ARM joint orientation sub-matrices and the Desired (given) target orientation sub-matrix: